# Maximizing the Spread of Influence through a Social Network

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9 A Conjecture. Finally, we state an appealing conjecture that would include all the approximability results above as special cases. CONJECTURE 4.3. Whenever the threshold functions f v at every node are monotone and submodular, the resulting influence function σ( ) is monotone and submodular as well. It is not difficult to show that every instance of the Triggering Model has an equivalent formulation with submodular node thresholds. Every instance of the Decreasing Cascade Model has such an equivalent formulation as well; in fact, the Decreasing Cascade condition stands as a very natural special case of the conjecture, given that it too is based on a type of diminishing returns. When translated into the language of threshold functions, we find that the Decreasing Cascade condition corresponds to the following natural requirement: f v(s {u}) f v(s) 1 f v(s) fv(t {u}) fv(t ), 1 f v(t ) whenever S T and u / T. This is in a sense a normalized submodularity property; it is stronger than submodularity, which would consist of the same inequality on just the numerators. (Note that by monotonicity, the denominator on the left is larger.) 5. NON-PROGRESSIVE PROCESSES We have thus far been concerned with the progressive case, in which nodes only go from inactivity to activity, but not vice versa. The non-progressive case, in which nodes can switch in both directions, can in fact be reduced to the progressive case. The non-progressive threshold process is analogous to the progressive model, except that at each step t, each node v chooses a new value θ v (t) uniformly at random from the interval [0, 1]. Node v will be active in step t if f v(s) θ v (t), where S is the set of neighbors of v that are active in step t 1. From the perspective of influence maximization, we can ask the following question. Suppose we have a non-progressive model that is going to run for τ steps, and during this process, we are allowed to make up to k interventions: for a particular node v, at a particular time t τ, we can target v for activation at time t. (v itself may quickly de-activate, but we hope to create a large ripple effect. ) Which k interventions should we perform? Simple examples show that to maximize influence, one should not necessarily perform all k interventions at time 0; e.g., G may not even have k nodes. Let A be a set of k interventions. The influence of these k interventions σ(a) is the sum, over all nodes v, of the number of time steps that v is active. The influence maximization problem in the non-progressive threshold model is to find the k interventions with maximum influence. We can show that the non-progressive influence maximization problem reduces to the progressive case in a different graph. Given a graph G = (V, E) and a time limit τ, we build a layered graph G τ on τ V nodes: there is a copy v t for each node v in G, and each time-step t τ. We connect each node in this graph with its neighbors in G indexed by the previous time step. THEOREM 5.1. The non-progressive influence maximization problem on G over a time horizon τ is equivalent to the progressive influence maximization problem on the layered graph G τ. Node v is active at time t in the non-progressive process if and only if v t is activated in the progressive process. Thus, models where we have approximation algorithms for the progressive case carry over. Theorem 5.1 also implies approximation results for certain non-progressive models used by Asavathiratham et al. to model cascading failures in power grids [2, 3]. Note that the non-progressive model discussed here differs from the model of Domingos and Richardson [10, 26] in two ways. We are concerned with the sum over all time steps t τ of the expected number of active nodes at time t, for a given a time limit τ, while [10, 26] study the limit of this process: the expected number of nodes active at time t as t goes to infinity. Further, we consider interventions for a particular node v, at a particular time t τ, while the interventions considered by [10, 26] permanently affect the activation probability function of the targeted nodes. 6. GENERAL MARKETING STRATEGIES In the formulation of the problem, we have so far assumed that for one unit of budget, we can deterministically target any node v for activation. This is clearly a highly simplified view. In a more realistic scenario, we may have a number m of different marketing actions M i available, each of which may affect some subset of nodes by increasing their probabilities of becoming active, without necessarily making them active deterministically. The more we spend on any one action the stronger its effect will be; however, different nodes may respond to marketing actions in different ways, In a general model, we choose investments x i into marketing actions M i, such that the total investments do not exceed the budget. A marketing strategy is then an m-dimensional vector x of investments. The probability that node v will become active is determined by the strategy, and denoted by h v(x). We assume that this function is non-decreasing and satisfies the following diminishing returns property for all x y and a 0 (where we write x y or a 0 to denote that the inequalities hold in all coordinates): h v(x + a) h v(x) h v(y + a) h v(y) (1) Intuitively, Inequality (1) states that any marketing action is more effective when the targeted individual is less marketing-saturated at that point. We are trying to maximize the expected size of the final active set. As a function of the marketing strategy x, each node v becomes active independently with probability h v(x), resulting in a (random) set of initial active nodes A. Given the initial set A, the expected size of the final active set is σ(a). The expected revenue of the marketing strategy x is therefore g(x) = A V σ(a) u A hu(x) v / A (1 hv(x)). In order to (approximately) maximize g, we assume that we can evaluate the function at any point x approximately, and find a direction i with approximately maximal gradient. Specifically, let e i denote the unit vector along the i th coordinate axis, and δ be some constant. We assume that there exists some γ 1 such that we can find an i with g(x + δ e i) g(x) γ (g(x + δ e j) g(x)) for each j. We divide each unit of the total budget k into equal parts of size δ. Starting with an all-0 investment, we perform an approximate gradient ascent, by repeatedly (a total of k times) adding δ δ units of budget to the investment in the action M i that approximately maximizes the gradient. The proof that this algorithm gives a good approximation consists of two steps. First, we show that the function g we are trying to optimize is non-negative, non-decreasing, and satisfies the diminishing returns condition (1). Second, we show that the hillclimbing algorithm gives a constant-factor approximation for any function g with these properties. The latter part is captured by the following theorem. THEOREM 6.1. When the hill-climbing algorithm finishes with strategy x, it guarantees that g(x) (1 e k γ k+δ n ) g(ˆx), where ˆx denotes the optimal solution subject to i ˆxi k.

10 The proof of this theorem builds on the analysis used by Nemhauser et al. [23], and we defer it to the full version of this paper. With Theorem 6.1 in hand, it remains to show that g is nonnegative, monotone, and satisfies condition (1). The first two are clear, so we only sketch the proof of the third. Fix an arbitary ordering of vertices. We then use the fact that for any a i, b i, a i b i = (a i b i) a j b j, (2) i i j<i j>i i and change the order of summation, to rewrite the difference g(x + a) g(x) = ( (hu(x + a) h u(x)) (σ(a + u) σ(a)) u A:u / A h j(x + a) (1 h j(x + a)) j<u,j A j>u,j A h j(x) j>u,j / A j<u,j / A (1 h j(x)) ). To show that this difference is non-increasing, we consider y x. From the diminishing returns property of h u( ), we obtain that h u(x + a) h u(x) h u(y + a) h u(y). Then, applying again equation (2), changing the order of summation, and performing some tedious calculations, writing (v, x, y) = h v(x + a) h v(y + a) if v < u, and (v, x, y) = h v(x) h v(y) if v > u, we obtain that (g(x + a) g(x)) (g(y + a) g(y)) ( (h u(y + a) h u(y)) (v, x, y) u,v:u v A:u,v / A j<min(u,v),j A u<j<v,j A v<j<u,j A ( σ(a + {u, v}) σ(a + v) σ(a + u) + σ(a) ) h j(x) j>max(u,v),j A h j(x + a) h j(y + a) u<j<v,j / A h j(y) j<min(u,v),j / A v<j<u,j / A (1 h j(x)) j>max(u,v),j / A (1 h j(x + a)) (1 h j(y + a)) ) (1 h j(y)) In this expression, all terms are non-negative (by monotonicity of the h v( )), with the exception of σ(a + {u, v}) σ(a + u) σ(a + v) + σ(a), which is non-positive because σ is submodular. Hence, the above difference is always non-positive, so g satisfies the diminishing returns condition (1). 7. REFERENCES [1] R. Albert, H. Jeong, A. Barabasi. Error and attack tolerance of complex networks. Nature 406(2000), [2] C. Asavathiratham, S. Roy, B. 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