more general models of social contagion in Visual Studio .NET Printer USS Code 39 in Visual Studio .NET more general models of social contagion

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more general models of social contagion use .net framework qr code creator todevelop qr code for .net Console application distribute qr-codes for .NET d. In order for u to succeed after all the nodes in X have tried and failed, we must further have v gv (X {u}).

Hence we should de ne the incremental function pv (u, X) = gv (X {u}) gv (X) . 1 gv (X). Conversely , suppose that we have incremental functions pv . Then the probability v is not activated by a set of neighbors X = {u1 , u2 , . .

. , uk } is k (1 pv (ui , Xi 1 )), i=1 where we write Xi 1 = {u1 , . .

. , ui 1 }. Note that order-independence is crucial here, to ensure that this quantity is independent of the way in which we label the elements of X.

Hence we can de ne a threshold function gv by setting. gv (X) = 1 . (1 pv (ui , Xi 1 ).. This compl qr barcode for .NET etes the translations in both directions, and hence establishes the equivalence of the two models. Next we consider some special cases of the Cascade Model that will be of particular interest to us.

(Given the equivalence to the General Threshold Model, these could also be written in that framework, though not always as simply.). (i) First, it is easy to encode the notion that v will deterministically activate once it has k active neighbors: we simply de ne pv (u, X) = 0 if . X. = k 1, and pv (u, X) = 1 if X. = k 1. (ii) In contrast, the in uence of a node s neighbors exhibits diminishing returns if it attenuates as more and more people try and fail to in uence it. Thus, we say that a set of incremental functions pv exhibits diminishing returns if pv (u, X) pv (u, Y ) whenever X Y .

(iii) A particularly simple special case that exhibits diminishing returns is the Independent Cascade Model, in which u s in uence on v is independent of the set of nodes that have already tried and failed: pv (u, X) = puv for some parameter puv that depends only on u and v.. We will se Denso QR Bar Code for .NET e that the contrast between (i) and (ii) above will emerge as a particularly important qualitative distinction: whether the in uence of one s neighbors in the social network incorporates some notion of critical mass (as in (i)), with a crucial number of adopters needed for successful in uence; or whether the strength of in uence simply decreases steadily (as in (ii)) as one is exposed more and more to the new behavior. In the next section, we will discuss an algorithmic problem whose computational complexity is strongly affected by this distinction; and following that, we will discuss some recent empirical studies that seek to identify the two sides of this dichotomy in online in uence data.

Before this, we brie y discuss a useful way of translating between the progressive and nonprogressive versions of these cascade processes. Progressive vs. nonprogressive processes (redux).

The discussion in this section has been entirely in terms of progressive processes, where nodes switching from the old behavior A to the new behavior B never switch back. There is a useful construction that allows one to study the nonprogressive version of the process by translation to. cascading behavior in networks a progress VS .NET QR Code ISO/IEC18004 ive one on a different graph (Kempe et al., 2003).

As it is a very general construction, essentially independent of the particular in uence rules being used, we describe it at this level of generality. Given a graph G on which we have a non-progressive process that may run for up to T steps, we create a larger graph G built from T copies of G, labeled G1 , G2 , . .

. , GT . Now, let v i be the copy of node v in the graph Gi ; we construct edges (u i 1 , v i ) for each neighbor u of v.

As a result, the neighbors of v i in G are just the copies of v s neighbors that live in the previous time-step. In this way, we can de ne the same in uence rules on G , node-by-node, that we had in G, and study the non-progressive process in G as a progressive process in G : some copies of v in G will be an active, and other will not, re ecting precisely the time steps in which v was active in G..

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