{"title": "Maximizing Influence in an Ising Network: A Mean-Field Optimal Solution", "book": "Advances in Neural Information Processing Systems", "page_first": 2495, "page_last": 2503, "abstract": "Influence maximization in social networks has typically been studied in the context of contagion models and irreversible processes. In this paper, we consider an alternate model that treats individual opinions as spins in an Ising system at dynamic equilibrium. We formalize the \\textit{Ising influence maximization} problem, which has a natural physical interpretation as maximizing the magnetization given a budget of external magnetic field. Under the mean-field (MF) approximation, we present a gradient ascent algorithm that uses the susceptibility to efficiently calculate local maxima of the magnetization, and we develop a number of sufficient conditions for when the MF magnetization is concave and our algorithm converges to a global optimum. We apply our algorithm on random and real-world networks, demonstrating, remarkably, that the MF optimal external fields (i.e., the external fields which maximize the MF magnetization) exhibit a phase transition from focusing on high-degree individuals at high temperatures to focusing on low-degree individuals at low temperatures. We also establish a number of novel results about the structure of steady-states in the ferromagnetic MF Ising model on general graphs, which are of independent interest.", "full_text": "Maximizing In\ufb02uence in an Ising Network:\n\nA Mean-Field Optimal Solution\n\nDepartment of Physics and Astronomy\n\nDepartment of Electrical and Systems Engineering\n\nChristopher W. Lynn\n\nUniversity of Pennsylvania\nchlynn@sas.upenn.edu\n\nDaniel D. Lee\n\nUniversity of Pennsylvania\nddlee@seas.upenn.edu\n\nAbstract\n\nIn\ufb02uence maximization in social networks has typically been studied in the context\nof contagion models and irreversible processes. In this paper, we consider an\nalternate model that treats individual opinions as spins in an Ising system at dynamic\nequilibrium. We formalize the Ising in\ufb02uence maximization problem, which has a\nnatural physical interpretation as maximizing the magnetization given a budget of\nexternal magnetic \ufb01eld. Under the mean-\ufb01eld (MF) approximation, we present a\ngradient ascent algorithm that uses the susceptibility to ef\ufb01ciently calculate local\nmaxima of the magnetization, and we develop a number of suf\ufb01cient conditions\nfor when the MF magnetization is concave and our algorithm converges to a\nglobal optimum. We apply our algorithm on random and real-world networks,\ndemonstrating, remarkably, that the MF optimal external \ufb01elds (i.e., the external\n\ufb01elds which maximize the MF magnetization) shift from focusing on high-degree\nindividuals at high temperatures to focusing on low-degree individuals at low\ntemperatures. We also establish a number of novel results about the structure of\nsteady-states in the ferromagnetic MF Ising model on general graph topologies,\nwhich are of independent interest.\n\n1\n\nIntroduction\n\nWith the proliferation of online social networks, the problem of optimally in\ufb02uencing the opinions\nof individuals in a population has garnered tremendous attention [1\u20133]. The prevailing paradigm\ntreats marketing as a viral process, whereby the advertiser is given a budget of seed infections and\nchooses the subset of individuals to infect such that the spread of the ensuing contagion is maximized.\nThe development of algorithmic methods for in\ufb02uence maximization under the viral paradigm has\nbeen the subject of vigorous study, resulting in a number of ef\ufb01cient techniques for identifying\nmeaningful marketing strategies in real-world settings [4\u20136]. While the viral paradigm accurately\ndescribes out-of-equilibrium phenomena, such as the introduction of new ideas or products to a\nsystem, these models fail to capture reverberant opinion dynamics wherein repeated interactions\nbetween individuals in the network give rise to complex macroscopic opinion patterns, as, for example,\nis the case in the formation of political opinions [7\u201310]. In this context, rather than maximizing\nthe spread of a viral advertisement, the marketer is interested in optimally shifting the equilibrium\nopinions of individuals in the network.\nTo describe complex macroscopic opinion patterns resulting from repeated microscopic interactions,\nwe naturally employ the language of statistical mechanics, treating individual opinions as spins in an\nIsing system at dynamic equilibrium and modeling marketing as the addition of an external magnetic\n\ufb01eld. The resulting problem, which we call Ising in\ufb02uence maximization (IIM), has a natural physical\ninterpretation as maximizing the magnetization of an Ising system given a budget of external \ufb01eld.\nWhile a number of models have been proposed for describing reverberant opinion dynamics [11], our\n\n30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.\n\n\fuse of the Ising model follows a vibrant interdisciplinary literature [12, 13], and is closely related\nto models in game theory [14, 15] and sociophysics [16, 17]. Furthermore, complex Ising models\nhave found widespread use in machine learning, and our model is formally equivalent to a pair-wise\nMarkov random \ufb01eld or a Boltzmann machine [18\u201320].\nOur main contributions are as follows:\n\n1. We formalize the in\ufb02uence maximization problem in the context of the Ising model, which\nwe call the Ising in\ufb02uence maximization (IIM) problem. We also propose the mean-\ufb01eld\nIsing in\ufb02uence maximization (MF-IIM) problem as an approximation to IIM (Section 2).\n\n2. We \ufb01nd suf\ufb01cient conditions under which the MF-IIM objective is smooth and concave,\nand we present a gradient ascent algorithm that guarantees an \u0001-approximation to MF-IIM\n(Section 4).\n\n3. We present numerical simulations that probe the structure and performance of MF optimal\nmarketing strategies. We \ufb01nd that at high temperatures, it is optimal to focus in\ufb02uence on\nhigh-degree individuals, while at low temperatures, it is optimal to spread in\ufb02uence among\nlow-degree individuals (Sections 5 and 6).\n\n4. Throughout the paper we present a number of novel results concerning the structure of\nsteady-states in the ferromagnetic MF Ising model on general (weighted, directed) strongly-\nconnected graphs, which are of independent interest. We name two highlights:\n\u2022 The well-known pitchfork bifurcation structure for the ferromagnetic MF Ising model\non a lattice extends exactly to general strongly-connected graphs, and the critical\ntemperature is equal to the spectral radius of the adjacency matrix (Theorem 3).\n\u2022 There can exist at most one stable steady-state with non-negative (non-positive) com-\n\nponents, and it is smooth and concave (convex) in the external \ufb01eld (Theorem 4).\n\n2 The Ising in\ufb02uence maximization problem\nWe consider a weighted, directed social network consisting of a set of individuals N = {1, . . . , n},\neach of which is assigned an opinion \u03c3i \u2208 {\u00b11} that captures its current state. By analogy with the\nIsing model, we refer to \u03c3 = (\u03c3i) as a spin con\ufb01guration of the system. Individuals in the network\ninteract via a non-negative weighted coupling matrix J \u2208 Rn\u00d7n\u22650 , where Jij \u2265 0 represents the\namount of in\ufb02uence that individual j holds over the opinion of individual i, and the non-negativity of\nJ represents the assumption that opinions of neighboring individuals tend to align, known in physics\nas a ferromagnetic interaction. Each individual also interacts with forces external to the network via\nan external \ufb01eld h \u2208 Rn. For example, if the spins represent the political opinions of individuals in a\nsocial network, then Jij represents the in\ufb02uence that j holds over i\u2019s opinion and hi represents the\npolitical bias of node i due to external forces such as campaign advertisements and news articles.\nThe opinions of individuals in the network evolve according to asynchronous Glauber dynamics. At\neach time t, an individual i is selected uniformly at random and her opinion is updated in response to\nthe external \ufb01eld h and the opinions of others in the network \u03c3(t) by sampling from\n\nP (\u03c3i(t + 1) = 1|\u03c3(t)) =\n\n,\n\n(1)\n\n(cid:80)\n\ne\u03b2((cid:80)\ni((cid:80)\ni=\u00b11 e\u03b2\u03c3(cid:48)\n\u03c3(cid:48)\n\nj Jij \u03c3j (t)+hi)\n\nj Jij \u03c3j (t)+hi)\n\nrefer to the total expected opinion, M =(cid:80)\n\nwhere \u03b2 is the inverse temperature, which we refer to as the interaction strength, and unless otherwise\nspeci\ufb01ed, sums are assumed over N. Together, the quadruple (N, J, h, \u03b2) de\ufb01nes our system. We\ni (cid:104)\u03c3i(cid:105), as the magnetization, where (cid:104)\u00b7(cid:105) denotes an average\nover the dynamics in Eq. (1), and we often consider the magnetization as a function of the external\n\ufb01eld, denoted M (h). Another important concept is the susceptibility matrix, \u03c7ij = \u2202(cid:104)\u03c3i(cid:105)\n, which\nquanti\ufb01es the response of individual i to a change in the external \ufb01eld on node j.\nWe study the problem of maximizing the magnetization of an Ising system with respect to the external\n\ufb01eld. We assume that an external \ufb01eld h can be added to the system, subject to the constraints\ni hi \u2264 H, where H > 0 is the external \ufb01eld budget, and we denote the set of feasible\ni hi = H}. In general, we also assume that the system\n\nh \u2265 0 and(cid:80)\nexternal \ufb01elds by FH = {h \u2208 Rn : h \u2265 0,(cid:80)\n\nexperiences an initial external \ufb01eld b \u2208 Rn, which cannot be controlled.\n\n\u2202hj\n\n2\n\n\fDe\ufb01nition 1. (Ising in\ufb02uence maximization (IIM)) Given a system (N, J, b, \u03b2) and a budget H, \ufb01nd\na feasible external \ufb01eld h \u2208 FH that maximizes the magnetization; that is, \ufb01nd an optimal external\n\ufb01eld h\u2217 such that\n\nh\u2217 = arg max\nh\u2208FH\n\nM (b + h).\n\n(2)\n\nNotation. Unless otherwise speci\ufb01ed, bold symbols represent column vectors with the appropriate\nnumber of components, while non-bold symbols with subscripts represent individual components.\nWe often abuse notation and write relations such as m \u2265 0 to mean mi \u2265 0 for all components i.\n\n2.1 The mean-\ufb01eld approximation\n\nIn general, calculating expectations over the dynamics in Eq. (1) requires Monte-Carlo simulations\nor other numerical approximation techniques. To make analytic progress, we employ the variational\nmean-\ufb01eld approximation, which has roots in statistical physics and has long been used to tackle\ninference problems in Boltzmann machines and Markov random \ufb01elds [21\u201324]. The mean-\ufb01eld\napproximation replaces the intractable task of calculating exact averages over Eq. (1) with the\nproblem of solving the following set of self-consistency equations:\n\n\uf8ee\uf8f0\u03b2\n\n\uf8eb\uf8ed(cid:88)\n\nj\n\n\uf8f6\uf8f8\uf8f9\uf8fb ,\n\nmi = tanh\n\nJijmj + hi\n\n(3)\n\nfor all i \u2208 N, where mi approximates (cid:104)\u03c3i(cid:105). We refer to the right-hand side of Eq. (3) as the\nmean-\ufb01eld map, f (m) = tanh [\u03b2(Jm + h)], where tanh(\u00b7) is applied component-wise. In this\nway, a \ufb01xed point of the mean-\ufb01eld map is a solution to Eq. (3), which we call a steady-state.\nIn general, there may be many solutions to Eq. (3), and we denote by Mh the set of steady-states for\na system (N, J, h, \u03b2). We say that a steady-state m is stable if \u03c1(f(cid:48)(m)) < 1, where \u03c1(\u00b7) denotes\nthe spectral radius and\n\nf(cid:48)(m)ij =\nwhere D(m)ij = (1 \u2212 m2\nsteady-state m, the susceptibility has a particularly nice form:\n\n\u2202fi\n\u2202mj\ni )\u03b4ij. Furthermore, under the mean-\ufb01eld approximation, given a stable\n\n(4)\n\n(cid:1) Jij \u21d2 f(cid:48)(m) = \u03b2D(m)J,\n\n(cid:12)(cid:12)(cid:12)(cid:12)m\n\n= \u03b2(cid:0)1 \u2212 m2\n(cid:33)\n\ni\n\nJik\u03c7kj + \u03b4ij\n\n\u21d2 \u03c7M F = \u03b2 (I \u2212 \u03b2D(m)J)\n\n\u22121 D(m),\n\n(5)\n\nij = \u03b2(cid:0)1 \u2212 m2\n\ni\n\n\u03c7M F\n\n(cid:1)(cid:32)(cid:88)\n\nk\n\nwhere I is the n \u00d7 n identity matrix.\nFor the purpose of uniquely de\ufb01ning our objective, we optimistically choose to maximize the\nmaximum magnetization among the set of steady-states, de\ufb01ned by\n\nM M F (h) = max\nm\u2208Mh\n\nmi(h).\n\n(6)\n\n(cid:88)\n\ni\n\nWe note that the pessimistic framework of maximizing the minimum magnetization yields an equally\nvalid objective. We also note that simply choosing a steady-state to optimize does not yield a\nwell-de\ufb01ned objective since, as h increases, steady-states can pop in and out of existence.\nDe\ufb01nition 2. (Mean-\ufb01eld Ising in\ufb02uence maximization (MF-IIM)) Given a system (N, J, b, \u03b2) and a\nbudget H, \ufb01nd an optimal external \ufb01eld h\u2217 such that\n\nh\u2217 = arg max\nh\u2208FH\n\nM M F (b + h).\n\n(7)\n\n3 The structure of steady-states in the MF Ising model\n\nBefore proceeding further, we must prove an important result concerning the existence and structure\nof solutions to Eq. (3), for if there exists a system that does not admit a steady-state, then our objective\n\n3\n\n\fis ill-de\ufb01ned. Furthermore, if there exists a unique steady-state m, then M M F =(cid:80)\n\ni mi, and there\n\nis no ambiguity in our choice of objective.\nTheorem 3 establishes that every system admits a steady-state and that the well-known pitchfork\nbifurcation structure for steady-states of the ferromagnetic MF Ising model on a lattice extends exactly\nto general (weighted, directed) strongly-connected graphs. In particular, for any strongly-connected\ngraph described by J, there is a critical interaction strength \u03b2c below which there exists a unique\nand stable steady-state. For h = 0, as \u03b2 crosses \u03b2c from below, two new stable steady-states appear,\none with all-positive components and one with all-negative components. Interestingly, the critical\ninteraction strength is equal to the inverse of the spectral radius of J, denoted \u03b2c = 1/\u03c1(J).\nTheorem 3. Any system (N, J, h, \u03b2) exhibits a steady-state. Furthermore, if its network is strongly-\nconnected, then, for \u03b2 < \u03b2c, there exists a unique and stable steady-state. For h = 0, as \u03b2 crosses\n\u03b2c from below, the unique steady-state gives rise to two stable steady-states, one with all-positive\ncomponents and one with all-negative components.\nProof sketch. The existence of a steady-state follows directly by applying Brouwer\u2019s \ufb01xed-point\ntheorem to f. For \u03b2 < \u03b2c, it can be shown that f is a contraction mapping, and hence admits a unique\nand stable steady-state by Banach\u2019s \ufb01xed point theorem. For h = 0 and \u03b2 < \u03b2c, m = 0 is the unique\nsteady-state and f(cid:48)(m) = \u03b2J. Because J is strongly-connected, the Perron-Frobenius theorem\nguarantees a simple eigenvalue equal to \u03c1(J) and a corresponding all-positive eigenvector. Thus,\nwhen \u03b2 crosses 1/\u03c1(J) from below, the Perron-Frobenius eigenvalue of f(cid:48)(m) crosses 1 from below,\ngiving rise to a supercritical pitchfork bifurcation with two new stable steady-states corresponding to\nthe Perron-Frobenius eigenvector.\nRemark. Some of our results assume J is strongly-connected in order to use the Perron-Frobenius\ntheorem. We note that this assumption is not restrictive, since any graph can be ef\ufb01ciently decomposed\ninto strongly-connected components on which our results apply independently.\nTheorem 3 shows that the objective M M F (b + h) is well-de\ufb01ned. Furthermore, for \u03b2 < \u03b2c, Theorem\n3 guarantees a unique and stable steady-state m for all b + h. In this case, MF-IIM reduces to\ni mi, and because m is stable, M M F (b + h) is smooth for all h by the\nimplicit function theorem. Thus, for \u03b2 < \u03b2c, we can use standard gradient ascent techniques to\nef\ufb01ciently calculate locally-optimal solutions to MF-IIM. In general, M M F is not necessarily smooth\nin h since the topological structure of steady-states may change as h varies. However, in the next\nsection we show that if there exists a stable and entry-wise non-negative steady-state, and if J is\nstrongly-connected, then M M F (b + h) is both smooth and concave in h, regardless of the interaction\nstrength.\n\nmaximizing M M F =(cid:80)\n\n4 Suf\ufb01cient conditions for when MF-IIM is concave\n\nWe consider conditions for which MF-IIM is smooth and concave, and hence exactly solvable by\nef\ufb01cient techniques. The case under consideration is when J is strongly-connected and there exists a\nstable non-negative steady-state.\nTheorem 4. Let (N, J, b, \u03b2) describe a system with a strongly-connected graph for which there\ni mi(b + h),\nM M F (b + h) is smooth in h, and M M F (b + h) is concave in h for all h \u2208 FH.\nProof sketch. Our argument follows in three steps. We \ufb01rst show that m(b) is the unique stable\nnon-negative steady-state and that it attains the maximum total opinion among steady-states. This\ni mi(b). Furthermore, m(b) gives rise to a unique and smooth branch\ni mi(b + h) for\n\nexists a stable non-negative steady-state m(b). Then, for any H, M M F (b + h) =(cid:80)\nguarantees that M M F (b) =(cid:80)\nof stable non-negative steady-states for additional h, and hence M M F (b + h) =(cid:80)\n\nall h > 0. Finally, one can directly show that M M F (b + h) is concave in h.\nRemark. By arguments similar to those in Theorem 4, it can be shown that any stable non-positive\nsteady-state is unique, attains the minimum total opinion among steady-states, and is smooth and\nconvex for decreasing h.\nThe above result paints a signi\ufb01cantly simpli\ufb01ed picture of the MF-IIM problem when J is strongly-\nconnected and there exists a stable non-negative steady-state m(b). Given a budget H, for any\nfeasible marketing strategy h \u2208 FH, m(b + h) is the unique stable non-negative steady-state,\nattains the maximum total opinion among steady-states, and is smooth in h. Thus, the objective\n\n4\n\n\fAlgorithm 1: An \u0001-approximation to MF-IIM\nInput: System (N, J, b, \u03b2) for which there exists a stable non-negative steady-state, budget H,\nOutput: External \ufb01eld h that approximates a MF optimal external \ufb01eld h\u2217\nt = 0; h(0) \u2208 FH; \u03b1 \u2208 (0, 1\nrepeat\n\naccuracy parameter \u0001 > 0\n\nL ) ;\n\n=(cid:80)\n(cid:2)h(t) + \u03b1(cid:79)hM M F (b + h(t))(cid:3);\n\n(b + h(t));\n\ni \u03c7M F\n\nij\n\n\u2202M M F (b+h(t))\n\n\u2202hj\n\nh(t + 1) = PFH\nt++;\n\nuntil M M F (b + h\u2217) \u2212 M M F (b + h(t)) \u2264 \u0001;\nh = h(t);\n\nM M F (b + h) =(cid:80)\n\ni mi(b + h) is smooth, allowing us to write down a gradient ascent algorithm that\napproximates a local maximum. Furthermore, since M M F (b+h) is concave in h, any local maximum\nof M M F on FH is a global maximum, and we can apply ef\ufb01cient gradient ascent techniques to solve\nMF-IIM.\nOur algorithm, summarized in Algorithm 1, is initialized at a feasible external \ufb01eld. At each iteration,\nwe calculate the susceptibility of the system, namely \u2202M M F\n, and project this gradient\n\u2202hj\nonto FH (the projection operator PFH is well-de\ufb01ned since FH is convex). Stepping along the\ndirection of the projected gradient with step size \u03b1 \u2208 (0, 1\nL ), where L is a Lipschitz constant of\nM M F , Algorithm 1 converges to an \u0001-approximation to MF-IIM in O(1/\u0001) iterations [25].\n\n=(cid:80)\n\ni \u03c7M F\n\nij\n\n4.1 Suf\ufb01cient conditions for the existence of a stable non-negative steady-state\n\nIn the previous section we found that MF-IIM is ef\ufb01ciently solvable if there exists a stable non-\nnegative steady-state. While this assumption may seem restrictive, we show, to the contrary, that the\nappearance of a stable non-negative steady-state is a fairly general phenomenon. We \ufb01rst show, for J\nstrongly-connected, that the existence of a stable non-negative steady-state is robust to increases in h\nand that the existence of a stable positive steady-state is robust to increases in \u03b2.\nTheorem 5. Let (N, J, h, \u03b2) describe a system with a strongly-connected graph for which there\nexists a stable non-negative steady-state m. If m \u2265 0, then as h increases, m gives rise to a unique\nand smooth branch of stable non-negative steady-states. If m > 0, then as \u03b2 increases, m gives rise\nto a unique and smooth branch of stable positive steady-states.\nProof sketch. By the implicit function theorem, any stable steady-state can be locally de\ufb01ned as a\nfunction of both h and \u03b2. Using the susceptibility, one can directly show that any stable non-negative\nsteady-state remains stable and non-negative as h increases and that any stable positive steady-state\nremains stable and positive as \u03b2 increases.\nThe intuition behind Theorem 5 is that increasing the external \ufb01eld will never destroy a steady-state in\nwhich all of the opinions are already non-positive. Furthermore, as the interaction strength increases,\neach individual reacts more strongly to the positive in\ufb02uence of her neighbors, creating a positive\nfeedback loop that results in an even more positive magnetization. We conclude by showing for J\nstrongly-connected that if h \u2265 0, then there exists a stable non-negative steady-state.\nTheorem 6. Let (N, J, h, \u03b2) describe any system with a strongly-connected network. If h \u2265 0, then\nthere exists a stable non-negative steady-state.\nProof sketch. For h > 0 and \u03b2 < \u03b2c, it can be shown that the unique steady-state is positive, and\nhence Theorem 5 guarantees the result for all \u03b2(cid:48) > \u03b2. For h = 0, Theorem 3 provides the result.\nAll together, the results of this section provide a number of suf\ufb01cient conditions under which MF-IIM\nis exactly and ef\ufb01ciently solvable by Algorithm 1.\n\n5\n\n\f5 A shift in the structure of solutions to MF-IIM\n\nThe structure of solutions to MF-IIM is of fundamental theoretical and practical interest. We\ndemonstrate, remarkably, that solutions to MF-IIM shift from focusing on nodes of high degree at\nlow interaction strengths to focusing on nodes of low degree at high interaction strengths.\nConsider an Ising system described by (N, J, h, \u03b2) in the limit \u03b2 (cid:28) \u03b2c. To \ufb01rst-order in \u03b2, the\nself-consistency equations (3) take the form:\n\nm = \u03b2 (Jm + h) \u21d2 m = \u03b2(I \u2212 \u03b2J)\u22121h.\n\nSince \u03b2 < \u03b2c, we have \u03c1(\u03b2J) < 1, allowing us to expand (I \u2212 \u03b2J)\u22121 in a geometric series:\ndout\ni hi + O(\u03b23),\n\nm = \u03b2h + \u03b22Jh + O(\u03b23) \u21d2 M M F (h) = \u03b2\n\nhi + \u03b22(cid:88)\n\n(cid:88)\n\ni\n\ni\n\nj Jji is the out-degree of node i. Thus, for low interaction strengths, the MF\nwhere dout\nmagnetization is maximized by focusing the external \ufb01eld on the nodes of highest out-degree in the\nnetwork, independent of b and H.\nTo study the structure of solutions to MF-IIM at high interaction strengths, we make the simplifying\nassumptions that J is strongly-connected and b \u2265 0 so that Theorem 6 guarantees a stable non-\nnegative steady state m. For large \u03b2 and an additional external \ufb01eld h \u2208 FH, m takes the form\n\ni = (cid:80)\n\nmi \u2248 tanh\n\n\uf8ee\uf8f0\u03b2\n\uf8eb\uf8ed(cid:88)\ni =(cid:80)\n(cid:16)\nM M F (b + h) \u2248(cid:88)\n\nj\n\n1 \u2212 2e\u22122\u03b2(din\n\ni\n\nwhere i\u2217 = arg mini(din\nfor an external \ufb01eld budget H are given by:\nn \u2212 2e\u22122\u03b2(din\n\n(cid:16)\n\nh\u2217 = arg max\nh\u2208FH\n\ni\u2217 +h(0)\n\nJij + bi + hi\n\ni +bi+hi),\n\n\uf8f6\uf8f8\uf8f9\uf8fb \u2248 1 \u2212 2e\u22122\u03b2(din\ni +bi+hi)(cid:17) \u2248 n \u2212 2e\u22122\u03b2(din\ni\u2217 +hi\u2217 )(cid:17) \u2261 arg max\n(cid:0)din\n\nmin\n\nh\u2208FH\n\ni\n\n(8)\n\n(9)\n\n(10)\n\n(11)\n\nwhere din\n\nj Jij is the in-degree of node i. Thus, in the high-\u03b2 limit, we have:\n\ni + bi + hi). Thus, for high interaction strengths, the solutions to MF-IIM\n\ni\u2217 +h(0)\n\ni\u2217 +hi\u2217 ),\n\n(cid:1) .\n\ni + bi + hi\n\n(12)\n\nEq. (12) reveals that the high-\u03b2 solutions to MF-IIM focus on the nodes for which din\ni + bi + hi is\nsmallest. Thus, if b is uniform, the MF magnetization is maximized by focusing the external \ufb01eld on\nthe nodes of smallest in-degree in the network.\nWe emphasize the strength and novelty of the above results. In the context of reverberant opinion\ndynamics, the optimal control strategy has a highly non-trivial dependence on the strength of\ninteractions in the system, a feature not captured by viral models. Thus, when controlling a social\nsystem, accurately determining the strength of interactions is of critical importance.\n\n6 Numerical simulations\n\nWe present numerical experiments to probe the structure and performance of MF optimal external\n\ufb01elds. We verify that the solutions to MF-IIM undergo a shift from focusing on high-degree nodes at\nlow interaction strengths to focusing on low-degree nodes at high interaction strengths. We also \ufb01nd\nthat for suf\ufb01ciently high and low interaction strengths, the MF optimal external \ufb01eld achieves the\nmaximum exact magnetization, while admitting performance losses near \u03b2c. However, even at \u03b2c,\nwe demonstrate that solutions to MF-IIM signi\ufb01cantly outperform common node-selection heuristics\nbased on node degree and centrality.\nWe \ufb01rst consider an undirected hub-and-spoke network, shown in Figure 1, where Jij \u2208 {0, 1} and\nwe set b = 0 for simplicity. Since b \u2265 0, Algorithm 1 is guaranteed to achieve a globally optimal MF\nmagnetization. Furthermore, because the network is small, we can calculate exact solutions to IIM\nby brute force search. The left plot in Figure 1 compares the average degree of the MF and exact\noptimal external \ufb01elds over a range of temperatures for an external \ufb01eld budget H = 1, verifying\n\n6\n\n\fFigure 1: Left: A comparison of the structure of the MF and exact optimal external \ufb01elds,\ndenoted h\u2217\ncompared to h\u2217; i.e., M (h\u2217\nM F\n\nM F and h\u2217, in a hub-and-spoke network. Right: The relative performance of h\u2217\n\nM F )/M (h\u2217\n\nM F ), where M denotes the exact magnetization.\n\nFigure 2: Left: A stochastic block network consisting of a highly-connected community\n(Block 1) and a sparsely-connected community (Block2). Center: The solution to MF-IIM\nshifts from focusing on Block 1 to Block 2 as \u03b2 increases. Right: Even at \u03b2c, the MF solution\noutperforms common node-selection heuristics.\n\nthat the solution to MF-IIM shifts from focusing on high-degree nodes at low interaction strengths\nto low-degree nodes at high interaction strengths. Furthermore, we \ufb01nd that the shift in the MF\noptimal external \ufb01eld occurs near the critical interaction strength \u03b2c = .5. The performance of the\nMF optimal strategy (measured as the ratio of the magnetization achieved by the MF solution to that\nachieved by the exact solution) is shown in the right plot in Figure 1. For low and high interaction\nstrengths, the MF optimal external \ufb01eld achieves the maximum magnetization, while near \u03b2c, it incurs\nsigni\ufb01cant performance losses, a phenomenon well-studied in the literature [21].\nWe now consider a stochastic block network consisting of 100 nodes split into two blocks of 50\nnodes each, shown in Figure 2. An undirected edge of weight 1 is placed between each pair of\nnodes in Block 1 with probability .2, between each pair in Block 2 with probability .05, and between\nnodes in different blocks with probability .05, resulting in a highly-connected community (Block\n1) surrounded by a sparsely-connected community (Block 2). For b = 0 and H = 20, the center\nplot in Figure 2 demonstrates that the solution to MF-IIM shifts from focusing on Block 1 at low \u03b2\nto focusing on Block 2 at high \u03b2 and that the shift occurs near \u03b2c. The stochastic block network is\nsuf\ufb01ciently large that exact calculation of the optimal external \ufb01elds is infeasible. Thus, we resort\nto comparing the MF solutions with three node-selection heuristics: one that distributes the budget\nin amounts proportional to nodes\u2019 degrees, one that distributes the budget proportional to nodes\u2019\ncentralities (the inverse of a node\u2019s average shortest path length to all other nodes), and one that\ndistributes the budget randomly. The magnetizations are approximated via Monte Carlo simulations\nof the Glauber dynamics, and we consider the system at \u03b2 = \u03b2c to represent the worst-case scenario\nfor the MF optimal external \ufb01elds. The right plot in Figure 2 shows that, even at \u03b2c, the solutions to\nMF-IIM outperform common node-selection heuristics.\nWe consider a real-world collaboration network (Figure 3) composed of 904 individuals, where each\nedge is unweighted and represents the co-authorship of a paper on the arXiv [26]. We note that\nco-authorship networks are known to capture many of the key structural features of social networks\n\n7\n\n\fFigure 3: Left: A collaboration network of 904 physicists where each edge represents the\nco-authorship of a paper on the arXiv. Center: The solution to MF-IIM shifts from high- to low-\ndegree nodes as \u03b2 increases. Right: The MF solution out-performs common node-selection\nheuristics, even at \u03b2c.\n\n[27]. For b = 0 and H = 40, the center plot in Figure 3 illustrates the sharp shift in the solution\nto MF-IIM at \u03b2c = 0.05 from high- to low-degree nodes. Furthermore, the right plot in Figure\n3 compares the performance of the MF optimal external \ufb01eld with the node-selection heuristics\ndescribed above, where we again consider the system at \u03b2c as a worst-case scenario, demonstrating\nthat Algorithm 1 is scalable and performs well on real-world networks.\n\n7 Conclusions\n\nWe study in\ufb02uence maximization, one of the fundamental problems in network science, in the\ncontext of the Ising model, wherein repeated interactions between individuals give rise to complex\nmacroscopic patterns. The resulting problem, which we call Ising in\ufb02uence maximization, has a\nnatural physical interpretation as maximizing the magnetization of an Ising system given a budget\nof external magnetic \ufb01eld. Under the mean-\ufb01eld approximation, we develop a number of suf\ufb01cient\nconditions for when the problem is concave, and we provide a gradient ascent algorithm that uses the\nsusceptibility to ef\ufb01ciently calculate locally-optimal external \ufb01elds. Furthermore, we demonstrate\nthat the MF optimal external \ufb01elds shift from focusing on high-degree individuals at low interaction\nstrengths to focusing on low-degree individuals at high interaction strengths, a phenomenon not\nobserved in viral models. We apply our algorithm on random and real-world networks, numerically\ndemonstrating shifts in the solution structure and showing that our algorithm out-performs common\nnode-selection heuristics.\nIt would be interesting to study the exact Ising model on an undirected network, in which case the\nspin statistics are governed by the Boltzmann distribution. Using this elegant steady-state description,\none might be able to derive analytic results for the exact IIM problem. Our work establishes a fruitful\nconnection between in\ufb02uence maximization and statistical physics, paving the way for exciting\ncross-disciplinary research. For example, one could apply advanced mean-\ufb01eld techniques, such\nas those in [21], to generate ef\ufb01cient algorithms of increasing accuracy. Furthermore, because our\nmodel is equivalent to a Boltzmann machine, one could propose a framework for data-based in\ufb02uence\nmaximization based on well-known Boltzmann machine learning techniques.\nAcknowledgements. We thank Michael Kearns and Eric Horsley for enlightening discussions, and\nwe acknowledge support from the U.S. National Science Foundation, the Air Force Of\ufb01ce of Scienti\ufb01c\nResearch, and the Department of Transportation.\n\nReferences\n[1] P. Domingos and M. Richardson. Mining the network value of customers. KDD, pages 57\u201366,\n\n2001.\n\n[2] M. Richardson and P. Domingos. Mining knowledge-sharing sites for viral marketing. 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Mean \ufb01eld theory for sigmoid belief networks.\n\nJournal of arti\ufb01cial intelligence research, 4(1):61\u201376, 1996.\n\n[25] M. Teboulle. First order algorithms for convex minimization. IPAM, 2010. Tutorials.\n\n[26] J. Leskovec and A. Krevl. SNAP Datasets: Stanford large network dataset collection, June\n\n2014.\n\n[27] M. Newman. The structure of scienti\ufb01c collaboration networks. PNAS, 98, 2001.\n\n9\n\n\f", "award": [], "sourceid": 1301, "authors": [{"given_name": "Christopher", "family_name": "Lynn", "institution": "University of Pennsylvania"}, {"given_name": "Daniel", "family_name": "Lee", "institution": "University of Pennsylvania"}]}